Neville is yet to manage his side to victory in seven Liga matches - a run that has seen the club drop to 11th in the Spanish top-flight.
And after being held to a 1-1 draw by Las Palmas in the first leg of their Copa del Rey quarter-final, the former Manchester United defender revealed Parejo would be replaced by Paco Alcacer as skipper.
The England coach did insist, however, that the player agreed with his decision and it was no reflection of his performance on the pitch.
"Giving the captaincy to Alacacer is something I had already talked to Parejo about in recent days," he told reporters.
"I have no problems with doing it. If I had a problem with Parejo, he would not play.
"We feel it is better for him to play without the burden of the armband."
After their cup exploits, Neville goes in search of his first league win on Sunday when his side travel to La Coruna to face Deportivo.